Paper Review

Summary

“The primary goal of this study is to investigate, using simulated data, the extent to which this ‘natural’ interpretation is robust and can be used to guide modelling decisions.”

• Period hyperparameter, \(P\), matches stellar rotation period.
• Timescale hyperparameter, \(\ell\), matches stellar spot evolution time.
• Harmonic complexity hyperparameter, \(\Gamma\), has weak relationship with physical parameters.
• No practical difference found between QP and QPC kernel results.

Quasi-periodic (QP) kernel

• Very common for stellar activity modelling
• Radial velocities (RVs) are believed to be quasi-periodic
• QP hyperparameters seem to have a natural interpretation
  • rotation rates
  • lifetimes of active regions

Squared Exponential kernel

\[k_\mathrm{SE}(\tau; A, \ell) = A \exp\left\{-\frac{1}{2}\left( \frac{\tau}{\ell}\right)^2\right\}\]

Periodic kernel

\[k_P(\tau; A, \Gamma, P) = A \exp\left\{ -\Gamma \sin^2\left[\pi \left( \frac{\tau}{P}\right) \right] \right\}\]

Cosine kernel

\[k_\mathrm{C}(\tau; A, P) = A \cos\left[2\pi \left( \frac{\tau}{P}\right)\right]\]

Quasi-periodic + Cosine (QPC) kernel

• Add a cosine term with a period equal to half of that of QP sine-squared term
• Capture the signal at the first harmonic of the stellar rotation period (Perger et al. 2021)
  • a secondary peak in the autocorrelation function seen at lag equal to half the rotation period.

Four Studies

1. Degeneracy of QP hyperparameters
2. Light Curve models
3. Radial Velocity models
4. Effect of Downsampling

Study 1: Hyperparameter Degeneracy

• Investigate the stability of Gaussian Process fitting when using QP or QPC kernels.
• Basically can you recover the ground truth from simulated data.

Study 1: Methodology

1. Simulate light curves by drawing QP hyperparameters from priors.
2. Fit a Gaussian Process using QP kernel.
3. Check if the recovered hyperparameters match the one drawn from prior.
4. Repeat for QPC kernel.

Study 1: Results

Study 1: K-S Test?

Studies 2 & 3: Methodology

1. Simulate data from a model using known physical parameters.
2. Fit a Gaussian Process with QP kernel.
3. Compare estimated hyperparameters with known physical parameters.
4. Repeat with QPC kernel.

• Study 2: Light curves
• Study 3: Radial velocities

Software Used

• PySpot (Aigrain 2021) to simulate stellar spot light curve data.
• george (Ambikasaran et al. 2015) for implementing kernels.
• emcee (Foreman-Mackey et al. 2013) for MCMC..

Study 2: Model vs GP

Study 2: QP vs QPC

Study 3: Light Curves vs RVs

Study 3: QP vs QPC

Study 4: Effect of Downsampling

• RV observations are typically ground-based and therefore more sparsely sampled and have higher noise.
• Methodology
  1. Add noise to simulated RV data.
  2. Randomly sample 50%, 20%, 10%, and 6% of the data.
  3. Visually examine the differences.

Study 4: Results

Study 4: Results

Comments on Statistical Method

• Constant mean function (?)
• Uniform priors on log scale (?)
• Sophisticated burnin procedure (+)
• Thinning is good way to break serial correlation (+)
• Point estimate of median of posterior is used (!)
• K-S test to Normality of standardised residuals (!!)
• Graphical justification of conclusions (+)

Takeaways for Fitting Stellar Models

• Shortest gap in data points \(\lt\) shortest rotation period.
  • Rapidly rotating stars need large numbers of observations per season.
• Sparsity increases \(\rightarrow\) Timescale estimates decrease
• Undersampling means high frequency features cannot be resolved.
  • Good results when \(\tau \gg\) sampling interval.
• Recovery of \(\Gamma\) is strongly affected by noise in data.